Unemployment Rates Forecasts Unobserved Component Models versus SARIMA models

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1 Unemploymen Raes Forecass Unobserved Componen Models versus SARIMA models Barbara Będowska-Sójka 1 Absrac: In his paper we focus on he comparison of unemploymen raes forecasing accuracy using ime-varying parameer models and SARIMA models. We are paricularly ineresed in he forecass of he unemploymen rae of eigh Cenral and Easern European firs-wave accession counries: Esonia, Lavia, Lihuania, Czech Republic, Poland, Slovakia, Hungary and Slovenia wihin years. We use a rolling shor-erm forecas experimen in order o obain ou-of-sample es of forecas accuracy. Moreover, we examine also he dynamic asymmeries in unemploymen raes and he forecasing performance of differen models. We find ha he forecasing abiliy of he models depends no only on he forecasing horizon, bu also on he direcion of he movemen in unemploymen raes. The empirical evidence derived from our invesigaion suggess ha here is no bes single model, however SARIMA models alhough no including a cyclical componen end o perform beer han ohers for a longer forecass horizons. JEL classificaion codes: C, C53, E7; Keywords: unemploymen rae, unobserved componen, SARIMA models, forecasing accuracy 1 Deparmen of Economerics, Poznan Universiy of Economics, Al. Niepodległości 10, Poznań, Poland, Phone: , barbara.bedowska-sojka@ue.poznan.pl; 1

2 Inroducion An imporan quesion in forecasing of ime series is which model is he bes one. For more or less fory years ARMA-ype models have been used for modelling and forecasing economic ime series. This approach has a cerain feaure: all shocks, coming eiher from he cycle or from oher sources, are included in hese model s innovaions. Simulaneously in he las years he unobserved componen models seem o become very promising ool in forecasing differen economic series as i allows o separae ime series componens. In his paper we compare he forecasing accuracy of few unobserved componen models and few specificaions of seasonal auoregressive inegraed moving average (SARIMA) models. We are ineresed in comparison of differen models of unemploymen raes series in several Cenral and Easern European (CEE) counries.. Nefci [1984] indicae ha some macroeconomic series display asymmeric behaviour. In case of he unemploymen raes hey have a endency o rise suddenly, bu fall gradually (Koop and Poer 1999). In his paper we are ineresed if here are he differences of he unemploymen raes forecass accuracy a he ime of increase and decrease of hese raes. For he purpose of forecasing we use linear models: in case of srucural ime series modelling level, rend, seasonaliy and cyclical componens are included, allowing for he coefficiens on each predicor o be eiher ime variable or consan over ime. In he case of SARIMA models we consider wo differen specificaions. The forecasing performance of differen models is compared in differen horizons and differen imes in order o indicae he bes model. A number of research papers have used ime series models for forecasing unemploymen raes. These works are devoed eiher o single unemploymen rae, where clearly he mos popular is he US unemploymen rae (e.g. Mongomery e al. 1998, Alissimo and Violane 001, Caner and Hansen 001, Proiei 003, Koop and Poer 1999) or a comparison of models used in forecasing unemploymen raes from differen economies, eg. OECD counries (Skalin and Teräsvira 00, U.S., U.K., Canada, and Japan (Milas and Rohman 005), G7 counries (Teräsvira e al. 005) and he Balic Saes (Będowska-Sójka 015). Many works are devoed o comparison of differen models. Mongomery, Zarnowiz, Tsay and Tiao [1998] in a rolling forecass experimen for he US quarerly unemploymen

3 raes show ha non-linear models performed beer han he linear ARMA model in erms of forecasing errors when he unemploymen increased rapidly bu no elsewhere. Sock and Wason [1999] used a large daa se of U.S. macroeconomic ime series, including he monhly unemploymen rae, and showed ha linear models have beer forecasing accuracy han nonlinear ones. Opposiely, Teräsvira e al. [005] find ha he nonlinear LSTAR model urns ou o be beer han he linear or neural nework models when modelling unemploymen raes in G7 counries. Marcellino (00) generaed forecass of hree key economic variables: he growh rae of indusrial producion, he unemploymen rae and he inflaion. He showed ha bes forecas for indusrial producion was obained wihin linear models, whereas for he unemploymen rae he non-linear models generae beer forecass. Proiei [003] invesigaed he ou-of-sample performance of linear and nonlinear srucural ime series models of he seasonally adjused US unemploymen rae. Generally linear models are said o perform significanly beer han nonlinear models, bu a nonlinear specificaion ouperforms he seleced linear model a shor lead imes in periods of slowly decreasing unemploymen rae. The main purpose of his paper is o compare an accuracy of unemploymen rae forecas s obained from differen linear models, namely srucural ime series models and SARIMA models. Our approach is much in he same spiri of Proiei (003) and Będowska- Sójka (015) as i concenraes on he comparison of forecasing models on he basis of he shor-erm forecass. Wih respec o Proiei (003) paper we focus on seasonally unadjused daa from several counries and use linear srucural ime series models only. In Będowska- Sójka (015) few unobserved componen models are compared from he perspecive of forecass generaed for unemploymen raes of he hree Balic saes: Lihuania, Lavia and Esonia. In ha paper i is shown ha models which conain cyclical componens perform beer han oher unobserved componen models (Będowska-Sójka 015). Our sample daa consiss of seasonally unadjused monhly unemploymen raes of he eigh CEE counries ha joined European Union in 004 in he firs-wave accession. These counries are: Czech Republic, Esonia, Hungary, Lavia, Lihuania, Poland, Slovakia, and Slovenia. The sample sars in January 1999 and ends in March 015 (wih some excepions described below). The forecass of unemploymen raes are generaed from he rolling forecass experimen where seasonaliy effecs are buil direcly ino he forecasing procedure. In his paper we consider all forecass origin saring from January 008 and 3

4 ending in March 014. The forecass are se o horizons from one monh o one year. As he rolling window generally consiss of 108 observaions we obain 75 forecass for each series and each models. In order o compare forecass from differen models, we use common forecasing error measures. To he bes of our knowledge his is he firs sudy ha compares unemploymen rae forecass wihin hese eigh CEE counries. Our conribuion is as follows: firs in six ou of eigh cases seasonal ARIMA models offered beer forecasing accuracy han he unobserved componen models. Second, when comparing models across all counries in he sample, here are subsanial differences beween heir forecasing abiliies; he lowes mean percenage forecasing error for 1-monh horizon is 1.8% in case of Slovakian unemploymen rae and he highes is 8.67% for he Esonian one. In case of Esonian, Lavian and Slovenian unemploymen raes shocks ha increase unemploymen raes end o have greaer negaive impac on he model s forecasing abiliy han shocks ha lower unemploymen rae. Finally, he differences in forecasing errors obained from differen mehods are generally no serious. The plan of he paper is as follows. Nex secion describes he mehodology used in he empirical sudy. Then daa are presened and empirical resuls of he comparison of forecass are shown. In he las secion he conclusions are presened. Mehodology Our paper aims o compare forecass from wo alernaive specificaions ha are used o represen he dynamic properies of series, namely unobserved componen models (UC) and seasonal ARIMA models. When he disurbances are independen, idenically disribued and Gaussian, an ARIMA model wih resricions in he parameers is he reduced form of an unobserved componen model [Harvey 1989]. There is one aggregaed disurbance wihin he specificaion of ARIMA models, whereas unobserved componen models include componen disurbances. Thus, he laer may allow o discover he characerisics, ha are no observed in he reduced form of ARIMA model. In his paper we ry examine which of hese wo classes of he models is more appropriae when forecasing he unemploymen raes. The heory of srucural ime series models and ARIMA models is presened by Harvey [1989] below a shor presenaion of he models used in he sudy is given. Wihin ARIMA models we use wo specificaions: I. Seasonal ARIMA(0,1,1)(0,1,1) henceforh SARIMA1 4

5 II. Seasonal ARIMA(,1,0)(0,1,1) henceforh SARIMA. When unobserved componen models are aken ino accoun, he general srucural model is wrien as [Harvey 1989]: y ~ NID (0, ) 1,..., T (1) where y represens he ime series o be modelled and forecas, is he seasonal componen, componen and is he cyclical componen, is he rend componen, represens he irregular NID denoes Normally and Independenly Disribued. All of hese componens are assumed o be unobserved. We use hree specificaions of UC models: III. where y Basic Srucural Model (BSM) ~ NID(0, ) 1 1 ~ NID(0, ) () 1 represens he sochasic level of he rend and he rend. I is also assumed ha, and rigonomeric seasonal componen described as: s / j1 represens he sochasic slope of are independen variables. Addiionally, is (3) wih s sanding for he number of seasons, s = 1 in our case. Each is generaed by: * cos j sin j sin j cos j 1 * 1, * j 1,...,[ s / ], 1,..., T (4) where * j j / s is he frequency and,, he seasonal disurbances, are muually * uncorrelaed ( ~ NID (0, ), j ~ NID (0, )) and uncorrelaed wih. j, * j As he unemploymen rae ends o move in a counercyclical way [Mongomery e al. 1998], we expec ha a cyclical componen migh improve unemploymen raes forecass. Therefore he nex model is: IV. y Srucural Model Plus Cycle (SMC), 1 In he model he saisical specificaion of a cycle,, is defined by: 5

6 cos sin c c 1, * * * sin c cos c 1 1,... T (5) where: c is he frequency (in radians), 0 c, is a damping facor, 0 1 and are muually uncorrelaed whie noise disurbances wih zero means and common *, variance. The las model included in he sudy is: V. Auoregressive Srucural Model (ARSM) y y1, 1 ~ NID(0, ) (6) where represens he sochasic level of he rend, is rigonomeric seasonal componen described in equaions (3) and (4), represens he irregular componen (as in equaion 1). As he primary use of ime series models is forecasing, i seems ha mean square error MSE would be adequae crierion in judging he models performance [Mongomery e al. 1998]. We use ou-of-sample forecass o assess which model gives he beer accuracy. These forecass are generaed in a rolling forecass window: for he given origin he model is esimaed and forecass are generaed. Nex, his sep is repeaed for each model and each series hence we obain 75 forecass from one-sep ahead ill welve-sep ahead for each series. The only excepion is he series of unemploymen raes of Slovakia, where he daa sars in 006 in his case we roll he forecass one sep a a ime forward, each ime reesimaing he model by exending he esimaion window. Finally, for all series and forecass we calculae differen forecasing errors and idenify he models wih he lowes errors. We also divide whole forecass origin ino increases and decreases in unemploymen raes and examine if here are any differences beween forecasing errors in hese wo saes. Daa Our sample daa consiss of monhly unemploymen raes from eigh firs-wave accession Cenral and Easern European counries ha joined European Union in May 004. There are (in alphabeical order): Czech Republic (CZ), Esonia (EE), Hungary (HU), Lavia (LA), Lihuania (LIT), Poland (PL), Slovenia (SI) and Slovakia (SK). We consider logarihms of monhly seasonally unadjused series. The seasonaliy is included in he models: in he unobserved componen models seasonal componen is modelled as a sochasic one. 6

7 The daa source is CEIC daabase ( sample sars in January 1999 and ends in March 015 wih some minor excepions. The daa for Esonian unemploymen rae sars in 001, for Slovenia sars in 000, and for Slovakia in 006 (in all cases he firs monh of he available daa is January). In case of he series ha are available since January 1999 saring from ha dae each model is esimaed and forecass from 1 monh ill 1 monhs are compued. The process is repeaed unil he end of sample is reached. In case of Esonian and Slovenian unemploymen rae he pre-forecass period is exended unil i reaches 108 observaions and hen he rolling window procedure is applied. The experimen provides in oal 75 forecass for horizons from one-monh o one-year for each model and each series. In case of unemploymen rae of Slovakia pre-forecass period is exended each ime wih a new informaion ill March 014 when he las forecas are generaed. The forecass origin consiss of he period of more or less rapid increase in he unemploymen raes as well as he gradual decrease wha give he possibiliy of observing he forecasing accuracy in differen business cycle phases. Figure 1 Unemploymen raes in he firs-wave EU accession CEE counries wihin CZ sands for Czech Republic, EE for Esonia, HU for Hungary, LV for Lavia, LIT for Lihuania, PL for Poland, SI for Slovenia and SK for Slovakia. 7

8 The calculaions and graphics are done in OxMerics STAMP7 [Koopman, Harvey, Doornik and Shephard 006, Doornik and Hendry 005]. Figure 1 shows how unemploymen raes changes wihin he sample period. There is no single endency for he unemploymen raes in he region a ha ime, bu some common feaures are recognizable. A he beginning of he sample some unemploymen raes are increasing and some decreasing. Saring from 001 he unemploymen raes in he region are decreasing (wih Hungarian rae as he excepion). There is also a visible change in all series as hey sar o increase sharply in he beginning or he mid of 008 and sar o decrease in he mid 010 (wih he excepion of Slovenia). Zooming i he single series behave differenly, some having huge differences beween he lowes and he highes poin. In he whole sample he highes unemploymen rae was observed in Poland in March 003 and he lowes in Esonia in December 006. The common feaure is he dynamic asymmery which is observable in all series: he decrease in unemploymen raes is raher gradual, whereas he increase is very seep. Empirical resuls The comparaive performance of a rolling forecas experimen is presened in hree seps. In he firs one an ou-of-sample es of forecas accuracy for he whole forecass origin is shown. Then we compare he forecass errors in wo cases: increase and decrease in he series. In he hird sep, he errors are depiced ogeher wih he series in order o illusrae in which periods we observe he bigges and he lowes errors. We repor comparaive performance of he rolling forecass in he models used in he sudy and described earlier. Tables 1 presens he differen forecasing errors for each series whereby: ~ is he l-ahead forecas for a given model, he Mean Error (ME) is obained as y l an average of forecass errors, y ~ y l calculaed as square roo of averages of, he Mean Square Forecas Error (MSFE) is ~ ( y y l ), and he Mean Absolue Percenage Error, MAPE, is obained as an average of y ~ y / y *100%. These errors are repored for 1-monh and 1-year horizon. l 8

9 Table 1. Comparison of forecass performance in he es period for unemploymen raes CEE counries CZ EE HU LA 1 monh ME RMSE MAPE ME RMSE MAPE ME RMSE MAPE ME RMSE MAPE SARIMA SARIMA BSM SMC ARSM monhs SARIMA SARIMA BSM SMC ARSM LIT PL SI SK 1 monh ME RMSE MAPE ME RMSE MAPE ME RMSE MAPE ME RMSE MAPE SARIMA SARIMA BSM SMC ARM monhs SARIMA SARIMA BSM SMC ARSM The bolded values are he lowes in a given horizon. 9

10 In mos cases he lowes forecass errors are obained from he same model for 1 monh and 1 monhs horizon. The average difference beween forecasing errors from differen models are raher small. In erms of considered forecasing errors, he greaes accuracy is provided by one of he seasonal ARIMA models (for CZ, LA, PL, SI and SK for boh horizons, whereas for HU for 1 monh horizon). On aggregae he seasonal ARIMA models ouperform unobserved componen models. The empirical evidence speaks srongly agains BSM model as i is he only one which is ouperformed by oher models for all series. There is a rade-off beween a Mean Error and he Mean Square Forecas Error or Mean Absolue Percenage Error: he lowes forecass bias measured by Mean Error is observed for he models ha have higher forecass variabiliy. In he nex sep he forecass origin is divided ino wo subsamples depending on increase or decrease (or remaining a he same level) of unemploymen raes. The formerly described errors are calculaed separaely for hese wo saes. Table presens he resuls of - es of equaliy of wo sample means [Snedecor and Cochran 1989]. Table Two-sample -Tes for equal means of errors in ime of unemploymen raes increase or decrease CZ EE HU LA LIT PL SI SK 1 monh monhs Bolded values are saisically significan a significance level α = The saisics are presened for seasonal ARIMA(,1,0)(0,1,1) model and MAPE errors, however he resuls of he saisical inerference are no changed for oher models as well as for ME or MSFE. According o he numbers presened in Table, in case of Esonian, Lavian and Slovenian one-monh forecass of unemploymen raes, errors coming from he forecass generaed for he ime of increase in unemploymen raes are sysemaically higher han errors obained in case of decrease in unemploymen raes. This resul holds also for Esonian and Lavian 1-monh forecass. We also presen one-sep ahead Mean Absolue Percenage Error from SARIMA model for one monh and 1 monhs horizon in 75 consecuive periods in forecass origin ogeher wih he unemploymen series. Figure shows ha he forecasing accuracy scores beer in periods of gradual decrease or increase in unemploymen raes and deerioraes in he beginning of he periods of rapid increase or decrease in he series. Similar behavior, alhough no presened here, characerizes he MAPE of mulisep forecass and oher errors aken ino 10

11 accoun in he sudy (ME and MSFE). I is conrary o wha was found in Proiei (003) wih respec o US unemploymen rae. Figure Unemploymen raes and one monh MAPE for eigh firs-wave EU accession CEE counries wihin CZ sands for Czech Republic, EE for Esonia, HU for Hungary, LV for Lavia, LIT for Lihuania, PL for Poland, SI for Slovenia and SK for Slovakia. Conclusion In his paper we have examined he ou-of-sample performance of wo alernaive specificaions ha are used o represen he dynamic properies of ime series, namely linear models for unemploymen raes of eigh CEE counries ha have accessed European Union in May 004. As he main ineres is o selec he bes forecasing models according o heir pos-sample performance, we have used rolling forecass experimen and examine, which 11

12 model generaed he bes forecass. Saring in January 1999 and ending in March 015 our sample consiss of he periods of decrease and increase in unemploymen raes. We find ha for he monhly daa in majoriy of cases seasonal ARIMA models perform beer han unobserved componen models considered in he sudy. The forecasing abiliy across differen series is surprisingly differenial. Generally speaking ARIMA models prove o be a very useful forecasing ool, boh for 1 monh and 1 monhs horizon. Only for wo series in he sample, he Esonian and he Hungarian unemploymen raes, he srucural ime series models give beer forecass. When periods of increases and decreases in he unemploymen raes are considered separaely, forecasing errors for hese wo saes are significanly differen only in hree cases. Las bu no leas he forecasing accuracy deerioraes in periods of rapid upward and downward movemen and improves in periods of gradual change in he unemploymen raes. References: Alissimo, F. and G.L. Violane, 001. The Non-Linear Dynamics of Oupu and Unemploymen in he U.S., Journal of Applied Economerics 16: Będowska-Sójka, B., 015. Unemploymen Rae Forecass. The Evidence from he Balic Saes, Easern European Economics 53: Caner, M. and B.E. Hansen, 001. Threshold Auoregression wih a Uni Roo, Economerica 69: Doornik, J.A., and D.F. Hendry, 005. Empirical Economeric Modelling. PcGive TM 11, Timberlake Consulans, London. Harvey, A.C., Forecasing Srucural Time Series Models and he Kalman Filer, Cambridge: Cambridge Universiy Press. Koop, G. S. and M. Poer, Dynamic Asymmeries in U.S. Unemploymen, Journal of Business and Economic Saisics 17 (3): Koopman, S.J., A.C. Harvey, J.A. Doornik, and N. Shephard Srucural Time Series Analyser and Modeller and Predicor STAMP 7, Timberlake Consulans, London. Marcellino, M. 00, Insabiliy and non-lineariy in he EMU, Discussion Paper No. 331, Cenre for Economic Policy Research. Milas, C. and P.Rohman "Mulivariae STAR Unemploymen Rae Forecass," Economerics , EconWPA. 1

13 Mongomery, A. L., V, Zarnowiz, R. S. Tsay, and G.C. Tiao Forecasing he U.S. Unemploymen Rae, Journal of he American Saisical Associaion 93, no. 44: Nefci, S.N Are Economic Time Series Asymmeric Over he Business Cycle? Journal of Poliical Economy 9, Proiei, T., 003. Forecasing he US unemploymen rae, Compuaional Saisics and Daa Analysis 4: Skalin, J., and T. Teräsvira, 00. Modeling asymmeries and moving equilibria in unemploymen raes, Macroeconomic Dynamics 6: Snedecor, G.W., and W.G. Cochran, Saisical Mehods, Eighh Ediion, Iowa Sae Universiy Press. Sock, J.H., and M.W. Wason, "Business cycle flucuaions in us macroeconomic ime series, in: Taylor, J. B., M., Woodford (ed.)" Handbook of Macroeconomics, volume 1: 3-64, Elsevier. Teräsvira, T., D. van Dijk,, and M. C. Medeiros, 005, Smooh ransiion auoregressions, neural neworks, and linear models in forecasing macroeconomic ime series: A reexaminaion, Inernaional Journal of Forecasing 1,